# Number Theory (analytic and combinatorial number theory, algebraic number theory, arithmetic geometry)

As a general rule, in number theory we find a constant interplay of a plethora of techniques coming from different areas of mathematics such as algebra, geometry, analysis, combinatorics, probability, ergodic theory, etc.

As for the general rule, also the research lines of number theory at ICMAT are many. The group studies several aspects of analytic, algebraic and combinatorial number theory and arithmetic geometry. The main research lines are the following:

• the applications of harmonic analysis in the Euclidean and hyperbolic setting to analytic number theory.
• additive combinatorics: it is a relatively recent term coined to comprehend the advancements of combinatorial number theory in problems related to the addition of integers and, more generally, of subsets of abelian groups.
• classical conjectures in number theory like the equivariant Birch-Swinerton Dyer conjecture and its generalizations, like the equivariant Tamagawa number conjecture.
• the arithmetic study of toric varieties, modular curves and modular varieties as well as elliptic curves and abelian varieties.
• the development of Arakelov theory
• the computation of rational points in curves and varieties
• the connections among formal groups, generalized Riemann-Hurwitz-type zeta functions and number theory
• the relation between the theory of motives, higher K-theory and arithmetic geometry

Of course all these problems and topics are related and depend on each other. For instance harmonic analysis is heavily used in studying combinatorial problems, some developments in Arakelov theory have been applied to the study of modular varieties, which are used to tackle the Birch and Swinnerton-Dyer conjecture, which gives informations about rational points on abelian varieties.

A pivotal theme in many of these areas is the study of the modular varieties, modular forms and automorphic forms. Their common appearance is a feature of great interest in modern number theory: it has connected number theorists, who are studying them with a few points of view and many different goals.

The objects involved in our research have recently been found very useful because of the increasing number of applications in other fields such as theoretical physics and cryptography. Although not yet ubiquitous, these tools are proving to play an important role also outside mathematics and this capture the attention of our research group.